Calculus 2 : Fundamental Theorem of Calculus

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #1 : Fundamental Theorem Of Calculus

Evaluate :

 

Possible Answers:

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we have that . Thus,

Example Question #3 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Evaluate  when .

Possible Answers:

Correct answer:

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function.

Therefore .

Example Question #2 : Fundamental Theorem Of Calculus

Evaluate  when .

Possible Answers:

Correct answer:

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function . Therefore, .

Example Question #1 : Fundamental Theorem Of Calculus

Evaluate  when .

Possible Answers:

Correct answer:

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function . Therefore, .

Example Question #2 : Fundamental Theorem Of Calculus

Suppose we have the function

What is the derivative, ?

Possible Answers:

Correct answer:

Explanation:

We can view the function  as a function of , as so

where .

We can find the derivative of  using the chain rule:

where  can be found using the fundamental theorem of calculus:

So we get

Example Question #3 : Fundamental Theorem Of Calculus

If , what is ?

Possible Answers:

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we know that given a function .

Thus, 

Example Question #4 : Fundamental Theorem Of Calculus

If , what is ?

Possible Answers:

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we know that given a function .

Thus, 

Example Question #5 : Fundamental Theorem Of Calculus

If , what is ?

Possible Answers:

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we know that given a function,   .

Thus, 

Example Question #8 : Fundamental Theorem Of Calculus

Given 

, what is ?

Possible Answers:

None of the above.

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, for all functions  that are continuously defined on the interval  with  in  and for all functions  defined by by , we know that .

Thus, for 

.

Therefore, 

Example Question #9 : Fundamental Theorem Of Calculus

Given 

, what is ?

Possible Answers:

None of the above.

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus,  for all functions  that are continuously defined on the interval  with  in  and for all functions  defined by by , we know that .  

Given 

, then 

.

Therefore, 

.

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